Chapter 1

Find all values of \(x\) satisfying the given conditions. $$ y_{1}=6\left(\frac{2 x}{x-3}\right)^{2}, y_{2}=5\left(\frac{2 x}{x-3}\right), \text { and } y_{1} \text { exceeds } y_{2} \text { by } 6 $$

Solve each absolute value inequality. $$\left|2-\frac{x}{2}\right|-1 \leq 1$$

Solve equation. \(0.5(x+2)=0.1+3(0.1 x+0.3)\)

Will help you prepare for the material covered in the next section. $$\text{Rationalize the denominator: }\frac{7+4 \sqrt{2}}{2-5 \sqrt{2}}$$

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ 3 x^{2}-12 x+12=0 $$

Solve each equation. $$ \left|x^{2}+2 x-36\right|=12 $$

Use interval notation to represent all values of \(x\) satisfying the given conditions. \(y_{1}=\frac{x}{2}+3, y_{2}=\frac{x}{3}+\frac{5}{2},\) and \(y_{1} \leq y_{2}\).

Solve equation. \(4 x+13-\\{2 x-[4(x-3)-5]\\}=2(x-6)\)

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ 9-6 x+x^{2}=0 $$

Solve each equation. $$ \left|x^{2}+6 x+1\right|=8 $$

Use interval notation to represent all values of \(x\) satisfying the given conditions. \(y_{1}=\frac{2}{3}(6 x-9)+4, y_{2}=5 x+1,\) and \(y_{1}>y_{2}\).

Solve equation. \(-2\\{7-[4-2(1-x)+3]\\}=10-[4 x-2(x-3)]\)

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ 4 x^{2}-16=0 $$

Solve each equation. $$ x(x+1)^{3}-42(x+1)^{2}=0 $$

Use interval notation to represent all values of \(x\) satisfying the given conditions. \(y=1-(x+3)+2 x\) and \(y\) is at least 4.

The mathematical model $$p+\frac{x}{2}=37$$ describes the percentage of Americans who smoked cigarettes, \(p\), \(x\) years after \(1970 .\) Use this model to solve. a. Does the mathematical model underestimate or overestimate the percentage of American adults who smoked cigarettes in \(2010 ?\) By how much? b. Use the mathematical model to project the year when only \(7 \%\) of American adults will smoke cigarettes.

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ 3 x^{2}-27=0 $$

Solve each equation. $$x(x-2)^{3}-35(x-2)^{2}=0$$

Use interval notation to represent all values of \(x\) satisfying the given conditions. \(y=2 x-11+3(x+2)\) and \(y\) is at most 0.

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ x^{2}-6 x+13=0 $$

Solve each equation. If 5 times a number is decreased by \(4,\) the principal square root of this difference is 2 less than the number. Find the number \((\mathrm{s})\)

Use interval notation to represent all values of \(x\) satisfying the given conditions. \(y=|3 x-4|+2\) and \(y<8\).

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ x^{2}-4 x+29=0 $$

Solve each equation. $$ \text { Solve for } V: r=\sqrt{\frac{3 V}{\pi h}} $$

Use interval notation to represent all values of \(x\) satisfying the given conditions. \(y=|2 x-5|+1\) and \(y>9\).

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ x^{2}=4 x-7 $$

Solve each equation. $$ \text { Solve for } V: r=\sqrt{\frac{3 V}{\pi h}} $$

Use interval notation to represent all values of \(x\) satisfying the given conditions. \(y=7-\left|\frac{x}{2}+2\right|\) and \(y\) is at most 4.

Here are two mathematical models for the data shown by the graph. In each formula, C represents the cost \(x\) years after 1980 of what cost \(\$ 10,000\) in 1967 $$\begin{aligned}&\text { Model } 1 \quad C=1388 x+24,963\\\&\text { Model 2 } \quad C=3 x^{2}+1308 x+25,268\end{aligned}$$ Use these models to solve. Use model 1 to determine in which year the cost will be \(\$ 77,707\) for what \(\operatorname{cost} \$ 10,000\) in 1967

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ 5 x^{2}=2 x-3 $$

Solve each equation. $$ \text { Solve for } A: r=\sqrt{\frac{A}{4 \pi}} $$

Use interval notation to represent all values of \(x\) satisfying the given conditions. \(y=8-|5 x+3|\) and \(y\) is at least 6.

Here are two mathematical models for the data shown by the graph. In each formula, C represents the cost \(x\) years after 1980 of what cost \(\$ 10,000\) in 1967 $$\begin{aligned}&\text { Model } 1 \quad C=1388 x+24,963\\\&\text { Model 2 } \quad C=3 x^{2}+1308 x+25,268\end{aligned}$$ Use these models to solve. Use model 1 to determine in which year the cost will be \(\$ 80,483\) for what \(\operatorname{cost} \$ 10,000\) in 1967

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ 2 x^{2}-7 x=0 $$

List all numbers that must be excluded from the domain of each expression. $$\frac{|x-1|-3}{|x+2|-14}$$

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ 2 x^{2}+5 x=3 $$

List all numbers that must be excluded from the domain of each expression. $$\frac{x^{3}-2 x^{2}-9 x+18}{x^{3}+3 x^{2}-x-3}$$

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ \frac{1}{x}+\frac{1}{x+2}=\frac{1}{3} $$

A basketball player's hang time is the time spent in the air when shooting a basket. The formula $$t=\frac{\sqrt{d}}{2}$$ models hang time, \(t,\) in seconds, in terms of the vertical distance of a player's jump, \(d,\) in feet. (image cannot copy) When Michael Wilson of the Harlem Globetrotters slamdunked a basketball, his hang time for the shot was approximately 1.16 seconds. What was the vertical distance of his jump, rounded to the nearest tenth of a foot?

A company wants to increase the \(10 \%\) peroxide content of its product by adding pure peroxide (100\% peroxide). If \(x\) liters of pure peroxide are added to 500 liters of its \(10 \%\) solution, the concentration, \(C\), of the new mixture is given by $$C=\frac{x+0.1(500)}{x+500}.$$ How many liters of pure peroxide should be added to produce a new product that is \(28 \%\) peroxide?

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ \frac{1}{x}+\frac{1}{x+3}=\frac{1}{4} $$

A basketball player's hang time is the time spent in the air when shooting a basket. The formula $$t=\frac{\sqrt{d}}{2}$$ models hang time, \(t,\) in seconds, in terms of the vertical distance of a player's jump, \(d,\) in feet. (image cannot copy) If hang time for a shot by a professional basketball player is 0.85 second, what is the vertical distance of the jump, rounded to the nearest tenth of a foot?

Suppose that \(x\) liters of pure acid are added to 200 liters of a \(35 \%\) acid solution. a. Write a formula that gives the concentration, \(C,\) of the new mixture. (Hint: See Exercise \(105 .\) ) b. How many liters of pure acid should be added to produce a new mixture that is \(74 \%\) acid?

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ \frac{2 x}{x-3}+\frac{6}{x+3}=-\frac{28}{x^{2}-9} $$

When 3 times a number is subtracted from 4, the absolute value of the difference is at least 5. Use interval notation to express the set of all numbers that satisfy this condition.

What is a linear equation in one variable? Give an example of this type of equation.

Solve each equation in Exercises \(83-108\) by the method of your choice. $$ \frac{3}{x-3}+\frac{5}{x-4}=\frac{x^{2}-20}{x^{2}-7 x+12} $$

When 4 times a number is subtracted from 5, the absolute value of the difference is at most 13. Use interval notation to express the set of all numbers that satisfy this condition.

Suppose that you solve \(\frac{x}{5}-\frac{x}{2}=1\) by multiplying both sides by 20 rather than the least common denominator (namely, 10 ). Describe what happens. If you get the correct solution, why do you think we clear the equation of fractions by multiplying by the least common denominator?

Suppose you are an algebra teacher grading the following solution on an examination: $$\begin{aligned}-3(x-6) &=2-x \\\\-3 x-18 &=2-x \\\\-2 x-18 &=2 \\\\-2 x &=-16 \\\x &=8\end{aligned}$$ You should note that 8 checks, so the solution set is \(\\{8\\}\) The student who worked the problem therefore wants full credit. Can you find any errors in the solution? If full credit is 10 points, how many points should you give the student? Justify your position.